An episodic history of mathematics. Mathematical culture through problem solving

Book Details

Author	Steven G. Krantz
Publisher	Mathematical Association of America
Year	2010
Language	English
Pages	381
Size	1.32 MB
Extension	PDF

Summary

An Episodic History of Mathematics: Mathematical Culture Through Problem Solving by Steven G. Krantz is a captivating journey through the development of mathematical thought, from the ancient Greeks to the twentieth century[reference:0][reference:1]. Published as part of the esteemed MAA Textbooks series by the Mathematical Association of America, this volume is not an encyclopedic chronicle but rather a series of vivid snapshots that bring mathematical culture and history to life[reference:2][reference:3]. Krantz's approach is "unabashedly mathematical," immersing the reader in the actual practice of mathematics through extensive examples and exercises, ensuring that the history is not merely observed but actively experienced[reference:4].

The book is structured chronologically across twenty-two chapters, each focusing on a pivotal figure, a transformative idea, or a defining period in mathematics[reference:5][reference:6]. It opens with the foundational work of Pythagoras, Euclid, and Archimedes, and concludes with a biographical sketch of Alan Turing and an introduction to his groundbreaking work on cryptography and Turing machines[reference:7]. In between, readers encounter the paradoxes of Zeno, the mystical mathematics of Hypatia, the algebraic innovations of the Islamic world, and the dramatic saga of solving polynomial equations featuring Cardano, Abel, and Galois[reference:8]. The narrative then proceeds through the coordinate geometry of Descartes, the invention of calculus, the profound contributions of Newton, Gauss, and Riemann, and the revolutionary set theory of Georg Cantor[reference:9]. The book also highlights the often-underrepresented contributions of women in mathematics with dedicated chapters on Sophie Germain, Sonya Kovalevskaya, and Emmy Noether[reference:10].

A key theme is the genesis and evolution of mathematical ideas. Krantz skillfully illustrates how problems and concepts arose, were challenged, and were eventually resolved or transformed[reference:11]. For instance, the reader is given a sense of the stream of thought that led to Fermat's Last Theorem, even without delving into the intricate details of its eventual proof by Andrew Wiles[reference:12]. This focus on the "why" and "how" behind the mathematics makes the subject both exciting and intellectually rewarding[reference:13]. The book abounds with stories and anecdotes, ensuring that the personalities behind the theorems are as memorable as the theorems themselves[reference:14].

The pedagogical approach is a defining strength of this work. Every chapter concludes with a detailed problem set, providing students with numerous avenues for exploration and reinforcing the concepts introduced[reference:15][reference:16]. Many exercises are designed to have the reader replicate the methods of historical figures, such as using Cardano's technique for solving cubic equations[reference:17]. This "learning by doing" philosophy transforms the reader from a passive observer into an active participant in the mathematical narrative[reference:18]. The book is therefore not just a history but a practical textbook that develops mathematical ability through engagement[reference:19].

Academically, An Episodic History of Mathematics holds significant importance as a bridge between historical context and mathematical practice. It is ideal for undergraduate courses in the history of mathematics, senior capstone courses, and for future mathematics teachers[reference:20][reference:21]. The book's approach addresses a common gap in mathematics education: the lack of historical perspective[reference:22]. By situating mathematical concepts within their historical and cultural context, Krantz provides a richer, more meaningful understanding of the discipline.

The target audience is broad, ranging from advanced high school students to university undergraduates and even graduate students seeking a broader perspective[reference:23]. It is also a valuable resource for mathematics teachers who wish to incorporate history into their curriculum[reference:24]. The book is accessible, yet it does not shy away from mathematical rigor. The recommended academic level is undergraduate, though the engaging narrative makes it appealing to anyone with a curiosity about the origins of mathematical ideas.

In summary, An Episodic History of Mathematics is a compelling and highly useful book. It succeeds in making the history of mathematics accessible, engaging, and deeply connected to the practice of problem-solving. Its strengths lie in its episodic structure, its focus on personalities and stories, and its commitment to involving the reader in the mathematics itself. It is an essential addition to any university or college library and a recommended read for all teachers and students of mathematics[reference:25].

Key Features

• Episodic and Chronological Structure: The book is organized into 22 self-contained chapters, each focusing on a specific historical figure, concept, or period, making it easy to digest and navigate[reference:26].
• Focus on Problem Solving: True to its title, the book emphasizes "doing" mathematics. Each chapter is followed by a substantial problem set that encourages active engagement and reinforces learning[reference:27].
• Rich with Biographical Anecdotes: The history is brought to life through engaging stories and personal details about the mathematicians, making the subject more relatable and memorable[reference:28].
• Broad Historical Coverage: It covers a wide span of mathematical history, from ancient Greek geometry to 20th-century cryptography, providing a comprehensive overview of the field's development[reference:29].
• Unabashedly Mathematical Approach: The book uses modern mathematical notation and includes numerous worked examples, ensuring that the historical narrative is grounded in actual mathematical practice[reference:30].
• Inclusive Representation: Dedicated chapters on Hypatia, Sophie Germain, Sonya Kovalevskaya, and Emmy Noether highlight the significant contributions of women in mathematics[reference:31].
• Excellent Pedagogical Tool: The structure, with its clear learning objectives and exercises, makes it an ideal textbook for undergraduate history of mathematics courses or senior capstone projects[reference:32].
• Balanced and Manageable Scope: Rather than attempting an exhaustive encyclopedic history, Krantz selects key episodes that illustrate the major themes and breakthroughs, making the subject accessible without sacrificing depth[reference:33].
• Valuable for Future Teachers: The book is highly recommended for prospective mathematics teachers, as it provides a historical perspective that can enrich their future teaching[reference:34].
• Engaging Writing Style: Krantz's prose is clear, engaging, and often lively, making the book a pleasure to read for both students and general enthusiasts[reference:35].

About Author

Steven G. Krantz is an American scholar, mathematician, and prolific writer[reference:36]. Born on February 3, 1951[reference:37], he earned his Bachelor's degree from the University of California, Santa Cruz in 1971, graduating summa cum laude[reference:38]. He went on to obtain his Ph.D. in mathematics from Princeton University in 1974 under the direction of Elias M. Stein and Joseph J. Kohn[reference:39].

Throughout his distinguished career, Krantz has taught at several prestigious institutions, including UCLA, Princeton University, Penn State University, and is currently a Professor of Mathematics at Washington University in St. Louis, where he has also served as Chair of his department[reference:40][reference:41]. He is a world-renowned expert in several complex variables, harmonic analysis, and partial differential equations[reference:42]. He was an inaugural Fellow of the American Mathematical Society in 2012[reference:43].

Krantz is an extraordinarily prolific author, having published over 150 books and more than 350 research papers[reference:44]. His contributions to mathematical literature span textbooks, research monographs, and expository works. He has been the recipient of numerous awards, including the prestigious Chauvenet Prize (1992), the Beckenbach Book Award, and the Kemper Foundation Award[reference:45][reference:46]. He has served as Editor-in-Chief of several notable journals, including the Notices of the American Mathematical Society and the Journal of Geometric Analysis[reference:47]. His work has had a profound impact on both the advancement of mathematics and the education of mathematicians.

Related Books

• A History of Mathematics – Carl B. Boyer and Uta C. Merzbach
• An Introduction to the History of Mathematics – Howard Eves
• The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities – William Dunham
• Journey Through Genius: The Great Theorems of Mathematics – William Dunham
• Mathematics and Its History – John Stillwell
• The History of Mathematics: An Introduction – David M. Burton
• Mathematical Thought from Ancient to Modern Times – Morris Kline
• God Created the Integers: The Mathematical Breakthroughs That Changed History – Stephen Hawking (Editor)

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Frequently Asked Questions

Q: What can readers learn from this book?

A: Readers will gain a deep appreciation for the history of mathematics and its cultural context. They will learn about the lives and work of key mathematicians, the genesis of important mathematical ideas, and will actively engage with mathematics through problem-solving exercises[reference:48].

Q: Who should read this book?

A: This book is ideal for undergraduate students in mathematics, future mathematics teachers, and anyone with a curiosity about the history and development of mathematical thought. It is also a valuable resource for professors designing courses in the history of mathematics[reference:49][reference:50].

Q: What background knowledge is required?

A: While the book is accessible, a basic understanding of high school and early undergraduate mathematics is recommended, as the text uses modern mathematical notation and includes challenging exercises[reference:51].

Q: Is this book suitable for students?

A: Yes, it is an excellent textbook for undergraduate courses. Its structure, with its clear chapters and problem sets, makes it ideal for both classroom use and independent study[reference:52].

Q: How does this book compare to other histories of mathematics?

A: Unlike comprehensive encyclopedic histories, this book takes an "episodic" approach, focusing on key moments and personalities. It is also more "mathematical" than many other histories, as it actively involves the reader in problem-solving, making it a more engaging and practical learning tool[reference:53][reference:54].

Q: What is the main strength of the book's pedagogical approach?

A: The main strength is its emphasis on "learning by doing." Every chapter ends with a detailed problem set that encourages students to engage directly with the mathematics, often by replicating the methods of historical figures[reference:55].

Q: Why is this book useful for future mathematics teachers?

A: It provides teachers with a rich historical context for the mathematics they will teach, offering stories and examples that can make the subject more engaging and meaningful for their own students[reference:56].